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Capacitance Lab PH 112 MJM February 6, 2007
You need your laptop for this experiment, so be sure to bring it. Plug in the USB cable before you
bring up Logger Pro. The collect button should be looking 'live' if all is ok. (If not, unplug and re-plug
the USB laptop connection, or try a different location to plug it in.)
The aluminum boxes each have a red push-button. You get no output unless you hold this button down!
You need to be able to open a new column of data for the RC discharge (the second part of the
experiment). This is explained on pp. 4-22 and 4-23. I didn't find that completely clear today.
What I did was
 Take voltage, time data for the RC discharge, and it plots the graph as it goes.
 Under the Data menu, I selected New Calculated Column
 Then I named the column ln(V), with a short name of lnV
 Then I clicked on the Functions button, and selected 'ln'
 Then I clicked on the little triangle at the right of the Variables (columns) space, and clicked on
'Potential', then clicked Done.
 To get a graph of ln(v) vs time I went to the Insert menu and clicked Graph, and the graph appeared.
 I dragged over straight part of the graph and clicked on the 'R=' tool on the top bar menu
 This gave the slope and stuff, but no standard deviation of the slope. I had to right-click and select
'Linear Fit Options' and tell it to display the standard deviation of the slope.
Make sure your fit information has at least 4 significant figures. If not, get the fit to give you 4 SF.
You should also open a 'meter' window, as explained on p. 4-18. This is extremely handy, to see what
the voltage probe is reading at all times.
I want you to do C1 and C2 in series as it calls for in the write-up, and I also want you to do C1 and
C2 in parallel, and check that they behave according to theory. You should derive two values for
C2, one from the series data and one from the parallel data. For capacitors in series we have
C = (C1 C2)/(C1+C2) and for capacitors in parallel we have C = C1+C2.
Rough check (to be done in addition to the other work.) The formula for decay is V = Vo exp(-t/(RC)).
The RC product is in seconds, since one ohm-farad = 1 second. e3 is almost exactly 20, so I want you
to do a rough check on your data using this idea. When t = 3 RC , or 3 time constants, the initial
voltage will have decayed to 1/20 of its initial value. Calculate what value of voltage equals 1/20 of the
initial voltage and use the 'x=' tool to move over the graph and find the time corresponding to this
voltage. The time interval between Vo and Vo/20 should be 3 RC, so please record this time, and
calculate a rough value of RC from it.
You don't need to save Excel-ready files if you do all the fits in Logger. And you don't need to do the
open-switch decay rate. But do save all the logger files in case you might need to retrieve them.
(over for some theory)
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Capacitance is defined as
Q = VC,
(1)
where Q is the charge on either plate
of the capacitor, C is the capacitance, and V is the potential difference across the plates of the cap.
When we take the time derivative of (1) we get
dQ/dt = C dV/dt
Since charge/time is current, dQ/dt = I = current, so
(2)
I = C dV/dt
(3)
When I is constant in the first part of the experiment, dV/dt is constant and V increases at a constant rate
across the capacitor. { From (3) you should be able to figure out the slope of the V vs. t line in terms of
I and C. }
I = -dQ/dt
In the second part of the experiment,
we discharge a capacitor across a resistor.
The circuit diagram is sketched at the
right. Notice that the charge Q on the
capacitor is decreasing, so I = -dQ/dt .
+
+Q
C
+
Vc = Q/C
-
R
-Q
I = -dQ/dt
V = Q/C, from (1), so the loop equation of the rises and drops in potential around this circuit is
+ Q/C - I R = 0 or
rise drop
Q/C = IR,
or
Q/C = -dQ/dt R
(4)
We rearrange (4) to put the Q's on one side at time t on the other.
dQ/Q = - dt/(RC)
(5)
The lefthand side of (5) is dimensionally charge/charge so it is a pure number. Then the righthand side
must also be dimensionally a pure number, time/ time. So RC has to be a time, and in fact
RC = ohms farads
and 1 ohm-farad equals 1 second.
When we integrate (5) we find
Q = Qo exp(-t/(RC))
(6)
Dividing both sides of (6) by C, and remembering that V = Q/C, we get
Q/C = Qo/C exp(-t/(RC))
or
V = Vo exp(-t/(RC))
Equation (7) is the same as Eq. (1) on p. 4-16 of the lab manual.
(7)